1 / 21

Fourier Transform and Applications

Fourier Transform and Applications. By Njegos Nincic. Fourier. Overview. Transforms Mathematical Introduction Fourier Transform Time-Space Domain and Frequency Domain Discret Fourier Transform Fast Fourier Transform Applications Summary References. Transforms.

Download Presentation

Fourier Transform and Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Fourier Transform and Applications By Njegos Nincic Fourier

  2. Overview • Transforms • Mathematical Introduction • Fourier Transform • Time-Space Domain and Frequency Domain • Discret Fourier Transform • Fast Fourier Transform • Applications • Summary • References

  3. Transforms • Transform: • In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica) • Can be thought of as a substitution

  4. Transforms • Example of a substitution: • Original equation: x + 4x² – 8 = 0 • Familiar form: ax² + bx + c = 0 • Let: y = x² • Solve for y • x = ±√y 4

  5. Transforms • Transforms are used in mathematics to solve differential equations: • Original equation: • Apply Laplace Transform: • Take inverse Transform: y = Lˉ¹(y)

  6. Fourier Transform • Property of transforms: • They convert a function from one domain to another with no loss of information • Fourier Transform: converts a function from the time (or spatial) domain to the frequency domain

  7. Time Domain and Frequency Domain • Time Domain: • Tells us how properties (air pressure in a sound function, for example) change over time: • Amplitude = 100 • Frequency = number of cycles in one second = 200 Hz

  8. Time Domain and Frequency Domain • Frequency domain: • Tells us how properties (amplitudes) change over frequencies:

  9. Time Domain and Frequency Domain • Example: • Human ears do not hear wave-like oscilations, but constant tone • Often it is easier to work in the frequency domain

  10. Time Domain and Frequency Domain • In 1807, Jean Baptiste Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions In picture: the composition of the first two functions gives the bottom one

  11. Time Domain and Frequency Domain

  12. Fourier Transform • Because of the property: • Fourier Transform takes us to the frequency domain:

  13. Discrete Fourier Transform • In practice, we often deal with discrete functions (digital signals, for example) • Discrete version of the Fourier Transform is much more useful in computer science: • O(n²) time complexity

  14. Fast Fourier Transform • Many techniques introduced that reduce computing time to O(n log n) • Most popular one: radix-2 decimation-in-time (DIT) FFT Cooley-Tukey algorithm: (Divide and conquer)

  15. Applications • In image processing: • Instead of time domain: spatial domain (normal image space) • frequency domain: space in which each image value at image position F represents the amount that the intensity values in image I vary over a specific distance related to F

  16. Applications: Frequency Domain In Images • If there is value 20 at the point that represents the frequency 0.1 (or 1 period every 10 pixels). This means that in the corresponding spatial domain image I the intensity values vary from dark to light and back to dark over a distance of 10 pixels, and that the contrast between the lightest and darkest is 40 gray levels

  17. Applications: Frequency Domain In Images • Spatial frequency of an image refers to the rate at which the pixel intensities change • In picture on right: • High frequences: • Near center • Low frequences: • Corners

  18. Applications: Image Filtering

  19. Other Applications of the DFT • Signal analysis • Sound filtering • Data compression • Partial differential equations • Multiplication of large integers

  20. Summary • Transforms: • Useful in mathematics (solving DE) • Fourier Transform: • Lets us easily switch between time-space domain and frequency domain so applicable in many other areas • Easy to pick out frequencies • Many applications

  21. References • Concepts and the frequency domain • http://www.spd.eee.strath.ac.uk/~interact/fourier/concepts.html • THE FREQUENCY DOMAIN Introduction • http://www.netnam.vn/unescocourse/computervision/91.htm • JPNM Physics Fourier Transform • http://www.med.harvard.edu/JPNM/physics/didactics/improc/intro/fourier2.html • Introduction to the Frequency Domain • http://zone.ni.com/devzone/conceptd.nsf/webmain/F814BEB1A040CDC68625684600508C88 • Fourier Transform Filtering Techniques • http://www.olympusmicro.com/primer/java/digitalimaging/processing/fouriertransform/index.html • Fourier Transform (Efunda) • http://www.efunda.com/math/fourier_transform/ • Integral Transforms • http://www.britannica.com/ebc/article?tocId=9368037&query=transform&ct=

More Related